In all these situations, although computations are certainly much more
difficult, it is important to underline that the basic ingredients
are the same: we continue to have a Kisin variety, together with a
shape stratification and we hope that those two objects strongly
govern the final form of the deformations space.
In full generality (i.e. for any reductive group, even not
necessarily \(\operatorname{GL}_n\) ), the Kisin variety and the shape stratification
are defined using techniques coming from group theory:
the Kisin variety is a subspace of the affine Grassmannian defined by
an explicit condition, which can be formulated in terms of the Cartan
decomposition, while the shape function \(x \mapsto g(x)\)  is defined by
means of the Iwahori-Cartan decomposition (and it now takes values in
the Iwahori-Weyl group).
After [1]}, we know moreover that Kisin varieties are related
with Pappas-Zhu local models [2]}; in this connection, the
shape stratification corresponds to the canonical stratification by
affine Schubert varieties.
